New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients

Pautrel, Thibault

doi:10.1214/20-ecp314

Cited by 2 publications

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References 15 publications

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“…Hence in the case of a spectral measure admitting a spectral density with respect to the Lebesgue measure, the expected number of real zeros has a whole spectrum of possible values. This has to be compared with the purely discrete case studied in [Thi20], where as recalled above, it is shown that, choosing µρ as purely atomic of the form µρ = 1 2 (δα + δ−α), with α ∈ πQ also yields non-universal nodal asymptotics ranging this time in the interval [ √ 2, 2].…”

Section: Main Results and Commentsmentioning

confidence: 99%

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On the real zeros of random trigonometric polynomials with dependent coefficients

Angst¹,

Dalmao²,

Poly³

2018

Proc. Amer. Math. Soc.

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We consider random trigonometric polynomials of the form f n (t) := 1≤k≤n a k cos(kt) + b k sin(kt), whose entries (a k ) k≥1 and (b k ) k≥1 are given by two independent stationary Gaussian processes with the same correlation function ρ. Under mild assumptions on the spectral function ψ ρ associated with ρ, we prove that the expectation of the number N n ([0, 2π]) of real roots of f n in the interval [0, 2π] satisfiesThe latter result not only covers the well-known situation of independent coefficients but allow us to deal with long range correlations. In particular it englobes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.

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Section: Main Results and Commentsmentioning

confidence: 99%

“…Recently in [Thi20], the author studied the case where the random coefficients still form a stationary Gaussian process, but the associated spectral measure in purely singular, namely ρ(k) = cos(kα) with α ∈ Q such that µρ = δα+δ −α 2…”

Section: Introductionmentioning

confidence: 99%

On the real zeros of random trigonometric polynomials with dependent coefficients

Angst¹,

Dalmao²,

Poly³

2018

Proc. Amer. Math. Soc.

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Real zeros of random trigonometric polynomials with dependent coefficients

Angst

Dalmao

Poly

2022

Trans. Amer. Math. Soc.

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We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form \[ f n ( t ) ≔ 1 n ∑ k = 1 n a k cos ⁡ ( k t ) + b k sin ⁡ ( k t ) , x ∈ [ 0 , 2 π ] , f_n(t)≔\frac {1}{\sqrt {n}}\sum _{k=1}^{n}a_k \cos (kt)+b_k\sin (kt), ~x\in [0,2\pi ], \] where the sequences ( a k ) k ≥ 1 (a_k)_{k\geq 1} and ( b k ) k ≥ 1 (b_k)_{k\geq 1} are two independent copies of a stationary Gaussian process centered with variance one and correlation function ρ \rho with associated spectral measure μ ρ \mu _{\rho } . We focus here on the case where μ ρ \mu _{\rho } is not purely singular and we denote by ψ ρ \psi _{\rho } its density component with respect to the Lebesgue measure λ \lambda . Quite surprisingly, we show that the asymptotics of the number of real zeros N ( f n , [ 0 , 2 π ] ) \mathcal {N}(f_n,[0,2\pi ]) of f n f_n in [ 0 , 2 π ] [0,2\pi ] is not related to the decay of the correlation function ρ \rho but instead to the Lebesgue measure of the vanishing locus of ψ ρ \psi _{\rho } . Namely, assuming that ψ ρ \psi _{\rho } is C 1 \mathcal {C}^1 with Hölder derivative on an open set of full measure, one establishes that \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = λ ( { ψ ρ = 0 } ) π 2 + 2 π − λ ( { ψ ρ = 0 } ) π 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {\lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {2}} + \frac {2\pi - \lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {3}}. \] On the other hand, assuming a sole log-integrability condition on ψ ρ \psi _{\rho } , which implies that it is positive almost everywhere, we recover the asymptotics of the independent case: \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = 2 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {2}{\sqrt {3}}. \] The latter asymptotics thus broadly generalizes the main result of Angst, Dalmao, and Poly [Proc. Amer. Math. Soc. 147 (2019), pp. 205–214] where the spectral density was assumed to be continuous and bounded from below. Besides, with further assumptions of regularity and existence of negative moment for ψ ρ \psi _{\rho } , which encompass e.g. the case of random coefficients being increments of fractional Brownian motion with any Hurst parameter, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.

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New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients

Cited by 2 publications

References 15 publications

On the real zeros of random trigonometric polynomials with dependent coefficients

On the real zeros of random trigonometric polynomials with dependent coefficients

Real zeros of random trigonometric polynomials with dependent coefficients

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